Book V. of A and C, M and G are equimultiples: And of B and D, N and K are equimultiples; if M be greater than N, G is greater bs.def. 5. than K; and if equal, equal; and if lefs, lefs; but G is greater than K, therefore M is greater than N: But H is not greater than L; and M, H are equimultiples of A, E; and N, L equimultiples of B, F: Therefore A has a greater ratio a 7. def. 5. 0 B, than E has to F2. Wherefore, if the firft, &c. Q E. D. COR. And if the firft has a greater ratio to the fecond, than the third has to the fourth, but the third the fame ratio to the fourth, which the fifth has to the fixth; it may be demonftrated in like manner that the first has a greater ratio to the second than the fifth has to the fixth. See N. 18. 5. IF ་ the first has to the fecond the fame ratio, which the third has to the fourth; then, if the first be greater than the third, the fecond fhall be greater than the fourth; and if equal, equal; and if less, less. Let the firft A have to the fecond B the fame ratio, which the third C has to the fourth D; if A be greater than C, B is greater than D. Because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C to B; But as A is to B, fo b 13.5. C IO. S. d 9. A B C D A B C D AB CD is C to D; therefore alfo C has to D a greater ratio than C has to Bb: But of two magnitudes, that to which the fame has the greater ratio is the leffer: Wherefore D is less than B; that is, B is greater than D. Secondly, If A be equal to C, B is equal to D: For A is to B, as C, that is A, to D; B therefore is equal to D Thirdly, If A be lefs than C, B fhall be less than D: For C is greater than A, and because C is to D, as A is to B, D is greater than B by the firft cafe; wherefore B is lefs than D. Therefore, if the firft, &c. Q. E. D. PROP, 'PROP. XV. THEOR. Book V. M AGNITUDES have the fame ratio to one another Let AB be the fame multiple of C, that DE is of F C is to F, as AB to DE. A D K Because AB is the fame multiple of C that DE is of F; there are as many magnitudes in AB equal to C, as there are in DE equal to F: Let AB be divided into magnitudes, each equal to C, viz AG, GH, HB; and DE into magnitudes, each equal to F, viz DK, KL. LE: Then the number of the first AG, GH, HB, fhall be equal to the number of the laft DK, KL, LE: And becaufe AG, GH, HB are Hall equal, and that DK, KL, LE are also equal to one another; therefore AG is to DK, as GH to KL, and as HB to LE2; And as one of the antecedents to its confequent; fo are all the antecedents together to L BCEF a 7. S. all the confequents together; wherefore, as AG is to DK, fo b 12. S. is AB to DE: But AG is equal to C, and DK to F: Therefore, as C is to F, fo is AB to DE. Therefore magnitudes, &c. Q. E. D. IF ly. four magnitudes of the fame kind be proportionals, they fhall alfo be proportionals when taken alternate Let the four magnitudes A, B, C, D be proportionals, viz. as A to B, fo C to D: They fhall alo be proportionals when taken alternately; that is, Á is to C, as B to D. Take of A and B any equimultiples whatever E and F ; and of C and D take any equimultiples whatever G and H And because Book V. becaufe E is the fame multiple of A, that F is of B, and that magnitudes have the fame ratio to one another which their equimultiples have; therefore A is to B, as E is to F; But as A is to B, fo is C to a 15. 5 b II. 5. € 14. 5. D: Wherefore, as CE is to D, fob is E to G C D H D, fo is E to F. Wherefore, as E is to F, fo is G to Hb. But, when four magnitudes are proportionals, if the first be greater than the third, the fecond fhall be greater than the fourth; and if equal, equal; if lefs, lefs. Wherefore, if E be greater than G, Flikewife is greater than H; and if equal, equal; if less, lefs: and E, F are any equimultiples whatever of A, B; and, G, H 5. def. 5. any whatever of C, D. Therefore A is to C, as B to Dd. If then four magnitudes, &c. QE. D. See N. a I. 5. IF magnitudes taken jointly be proportionals, they fhall alfo be proportionals when taken feparately; that is, if two magnitudes together have to one of them, the fame ratio which two others have to one of thefe, the remaining one of the first two fhall have to the other, the fame ratio which the remaining one of the last two has to the other of these. Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; that is, as AB to BE, fo is CD to DF; they fhall alfo be proportionals taken separately, viz. as AE to EB, fo CF to FD. Take of AE, EB, CF, FD any equimultiples whatever GH, HK, LM, MN; and again, of EB, FD, take any equimultiples whatever KX, NP: And because GH is the fame multiple of AE that HK is of EB, therefore GH is the fame multiple of AE, that GK is of AB: But GH is the fame multiple of AE, that LM is of CF; wherefore GK is the fame multiple of AB, that p b2.5. that LM is of CF. Again, because LM is the fame multiple of Book V. CF that MN is of FD; therefore LM is the fame multiple of CF, that LN is of CD: But LM was fhewn to be the fame a I. 5. multiple of CF, that GK is of AB; GK therefore is the fame multiple of AB, that LN is of CD; that is, GK, LN are equimultiples of AB, CD. Next, becaufe HK is the fame multiple of EB, that MN is of FD; and that KX is alfo the fame multiple of EB, that NP is of FD; therefore HX is the fame multiple of EB, that MP is of FD. And because AB is to BE, as CD is to DF, and that of AB and CD, GK and LN are equimultiples, and of EB and FD, HX and MP are K equimultiples; if GK be greater than HX, then LN is greater than MP; and if equal, equal; and if lefs, lefs: But if GH be H greater than KX, by adding the common part HK to both, GK is greater than HX; wherefore alfo LN is greater than MP; and by taking away MN from both, LM is greater than NP: Therefore, if GH be greater than KX, LM is greater than NP. E N c 5. def. 5 B DM F In like manner it may be demonftrated, GA CL that if GH be equal to KX, LM likewife is equal to NP; and if lefs, lefs: And GH, LM are any equimultiples whatever of AE, CF, and KX, NP are any whatever of EB, FD. Therefore, as AE is to EB, fo is CF to FD. If then magnitudes, &c. Q. E. D. IF magnitudes taken feparately be proportionals, they see N fhall alfo be proportionals when taken jointly, that is, if the first be to the fecond, as the third to the fourth, the first and fecond together fhall be to the fecond, as the third and fourth together to the fourth. Let AE, EB, CF, FD be proportionals; that is, as AE to EB, fo is CF to FD; they shall also be proportionals when taken jointly; that is, as AB to BE, fo CD to DF. Take of AB, BE, CD, DF any equimultiples whatever GH, HK, LM, MN; and again, of BE, DF, take any whatever equimultiples KO, NP: And because KO, NP are equimultiples of Book V. W of BE, DF; and that KH, NM are equimultiples likewife of a K Firft, Let KO not be greater than KH, therefore NP is not greater than NM: And because GH, HK are equimultiples of AB, BE, and that AB is greater than BE, therefore GH is a 3. Ax. greater than HK; but KO is not greater than KH, wherefore GH is greater than KO. In like manner it may be fhewn, that LM is greater than NP. Therefore, if KO be not greater than KH, then GH the multiple of AB is always greater than KO the GA multiple of BE; and likewife LM the bs. s. c 6. 5. B E F O L multiple of CD greater than NP the multiple of DF. Next, Let KO be greater than KH; therefore, as has been fhewn, NP is greater than NM: And because the whole GH is the fame multiple of the whole AB, that HK is of BE, the remainder GK is the fame multiple of H the remainder AE that GH is of AB", AE that LN is of CF; that is, GK, CL LN are equimultiples of AE, CF: GA And because KO, NP are equimul- there be taken KH, NM, which are likewife equimultiples that |